To find the derivative of the function $\frac{\sin x - \cos x}{\sin x + \cos x}$, we'll use the quotient rule. The quotient rule states that if you have a function of the form $\frac{u(x)}{v(x)}$, its derivative is given by:

$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

Here, let:

• $u(x) = \sin x - \cos x$
• $v(x) = \sin x + \cos x$

Step 1: Find the derivatives of $u(x)$ and $v(x)$

• $u'(x) = \cos x + \sin x$
• $v'(x) = \cos x - \sin x$

Step 2: Apply the quotient rule

Now, substitute $u(x)$, $v(x)$, $u'(x)$, and $v'(x)$ into the quotient rule formula:

$\frac{d}{dx}\left(\frac{\sin x - \cos x}{\sin x + \cos x}\right) = \frac{(\cos x + \sin x)(\sin x + \cos x) - (\sin x - \cos x)(\cos x - \sin x)}{(\sin x + \cos x)^2}$

Step 3: Simplify the numerator

Let's expand and simplify the numerator:

1. Expand $(\cos x + \sin x)(\sin x + \cos x)$: $(\cos x + \sin x)(\sin x + \cos x) = \cos^2 x + \cos x \sin x + \sin x \cos x + \sin^2 x = \cos^2 x + \sin^2 x + 2\cos x \sin x$ Since $\cos^2 x + \sin^2 x = 1$, the expression simplifies to: $1 + 2\cos x \sin x$

2. Expand $(\sin x - \cos x)(\cos x - \sin x)$: $(\sin x - \cos x)(\cos x - \sin x) = \sin^2 x - \sin x \cos x - \cos x \sin x + \cos^2 x = \sin^2 x + \cos^2 x - 2\sin x \cos x$ Again, $\sin^2 x + \cos^2 x = 1$, so this simplifies to: $1 - 2\sin x \cos x$

Thus, the numerator becomes: $(1 + 2\cos x \sin x) - (1 - 2\sin x \cos x) = 4\sin x \cos x$

Step 4: Simplify the expression

Finally, the derivative is: $\frac{4\sin x \cos x}{(\sin x + \cos x)^2}$

This can also be written using the identity $\sin 2x = 2\sin x \cos x$ as: $\frac{2\sin 2x}{(\sin x + \cos x)^2}$

Final Answer

The derivative of $\frac{\sin x - \cos x}{\sin x + \cos x}$ is:

$\frac{2\sin 2x}{(\sin x + \cos x)^2}$

Would you like any further details or explanations?

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Here are 5 related questions:

1. What is the derivative of $\frac{\sin x + \cos x}{\sin x - \cos x}$?
2. How would you find the second derivative of $\frac{\sin x - \cos x}{\sin x + \cos x}$?
3. Can the derivative $\frac{2\sin 2x}{(\sin x + \cos x)^2}$ be simplified further?
4. What is the integral of $\frac{\sin x - \cos x}{\sin x + \cos x}$?
5. How do you apply the quotient rule to functions involving trigonometric identities?

Tip: Remember that trigonometric identities can often simplify the differentiation or integration process, making complex expressions easier to handle.